Finite Rank Perturbations and Model Theory

许多物理系统都是用微分算子来建模的。描述系统长期行为的一种方法是通过频谱理论，其中包括发现系统自然表现出的频率。想象一根固定长度的振动弦或梁。显然，它的频率（想想“声音”）取决于它的末端（也就是边界）是如何被压制或限制的。一般来说，当施加在边界上的条件改变时，微分算子的谱以及与之相关的物理系统的性质会发生剧烈的变化。在许多情况下，我们知道一组边界条件的完整谱。从那里，我们可以通过所谓的有限阶摄动理论获得系统在其他条件下的知识。本课题的主要目标是系统地研究有限阶酉摄动的谱理论，并将其与相应的泛函模型联系起来。第一级的设置是相当容易理解的，而有限秩问题的一般处理提出了几个离题。最初，非循环酉非扰动算符提出了一个问题。这个问题现在解决了。一般问题涉及到矩阵值的赫氏变换和柯西变换。作为这个项目的一部分，这些变换的研究将提供对有限秩摄动理论的见解。计划对所谓的外部柯西变换进行正则化（就像在第一阶设置中所做的那样）。我们将进一步研究带缺陷算子的无穷阶摄动在迹类中的困难情况。类似的自伴随集，以及微分算子的具体应用，也将被研究。

Many physical systems are modeled by differential operators. One way of describing a system's long-term behavior is through spectral theory, which includes the finding of frequencies naturally exhibited by the system. Imagine a vibrating string or beam of fixed length. Clearly, its frequency (think "sound") depends on how its ends (aka boundaries) are clamped down or otherwise restricted. In general, the spectrum of a differential operator and with it the properties of a physical system can change drastically when the conditions imposed on the boundary are changed. In many cases, we know the complete spectrum for one set of boundary conditions. From there, we can gain knowledge about the system under other conditions via the theory of so-called finite rank perturbations. The primary goal of the project is to systematically study spectral theory of finite rank unitary perturbations by tightening the relationship to corresponding functional models. The rank one setting is reasonably well-understood, while a general treatment of the finite rank problem presented several digressions. Initially, non-cyclic unitary unperturbed operators posed an issue. This problem is now resolved. The general problem involves matrix-valued Herglotz and Cauchy-type transforms. The study of these transforms as part of this project will provide insight into finite rank perturbation theory. A regularization of the so-called exterior Cauchy transform (as was done in the rank one setting) is planned. Further investigations will be made into the difficult case of infinite rank perturbations with defect operators in the trace class. The analogous self-adjoint setting, as well as concrete applications to differential operators, will also be investigated.